Isometric properties of elementary operators
A bounded linear operator \(T\) on a complex Hilbert space \(H\) is an \(n\)-isometry if \[ \sum_{k=0}^{n}(-1)^{n-k}{ n\choose k} T^{*k}T^k=0. \] It is evident that an unitary operator is an \(n\)-isometry for every positive integer \(n\). Let \(S\) be a weighted shift on \(\ell^2\) induced by a bounded sequence \((\omega_k)_k\) of complex numbers. Then \(S\) is a \(2\)-isometry if and only if \[ |\omega_k|^2|\omega_{k+1}|^2-2|\omega_k|^2+1=0 \] holds for every \(k\geq 1\). It follows, for instance, that the weighted shift given by the sequence \(\omega_k=\sqrt{\frac{k+1}{k}}\) is a \(2\)-isometry which is not a \(1\)-isometry, i.e., an unitary operator. Weighted shifts that are \(3\)-isometries are characterized as well and it is shown in a new way that the multiplication operator with the independent variable on the Dirichlet space is a \(2\)-isometry. The second part of the paper deals with elementary operators of length \(1\) on the Hilbert-Schmidt class \({\mathcal C}_2(H)\). Let \(A\) and \(B\) be given bounded linear operators on \(H\) and let \(L\) be defined on \({\mathcal C}_2(H)\) by \(L(T)=ATB\). Then \(L\) is a \(2\)-isometry if and only if there exists a positive number \(\mu\) so that \(A^*A=\mu I\) and \(\sqrt{\mu}\, B^*\) is \(2\)-isometry, or \(BB^*=\mu I\) and \(\sqrt{\mu}\, A\) is \(2\)-isometry. A similar characterization is given for \(L\) to be a \(3\)-isometry. At the end, the \(n\)-isometry case is considered, however, only sufficient conditions for \(L\) to be an \(n\)-isometry are given.
scientific article; zbMATH DE number 5650045